TTT Discourse

Rules of a Tic Tac Toe universe

1. A statement is valid if it is made in order [rule 2] and if the square of the board indicated by the statement is un-owned [rule 3].
2. A statement made by a player is considered "in order" if the prior valid statement was made by the opposite player or if the player is player 1 and their statement is the first of the game.
3. When a player makes a valid statement, ownership of the square on the board coinciding with the statement is given to the player.
4. If a player owns three squares all in the same row, column, or diagonal of the board, that player has "won" the current game. The opposite player has "lost".
5. If all squares of the board are owned and neither player has won, the game is drawn. Neither player wins and neither player looses.
6. Once a winner is established or the game is drawn, ownership of all squares on the board is revoked returning each square to a neutral state. This starts a new game or iteration/generation of the universe.

I submit for consideration a reflection on an abstraction of the game of Tic-Tac-Toe.  This abstraction considers the moves of a tic-tac-toe game as a universe of discourse.  The rules of tic-tac-toe are, along the same line, the rules of the universe, governing the reaction of the universe to each "statement" made.

Three distinct entities make up the universe: two players and a board upon which the players interact.  The players I will refer to as player 1 and player 2 when differentiation is necessary.  The board is, to those who are familiar with the game, a standard tic-tac-toe board, consisting of 9 squares laid out in a 3x3 grid.  Players are the only acting entities.  Their interaction is performed via the board using the vocabulary set forth by the universe.  For purposes of the abstraction the specific semantics of the vocabulary are inconsequential so long as a player can formulate any valid tic-tac-toe move via use of the vocabulary.  The following set is one such vocabulary: { (-1,1), (0,1), (1,1), (-1,0), (0,0), (1,0), (-1,-1), (0,-1), (1,-1) } where each element in the set represents a square on the board.  Using a standard Cartesian coordinate system, the element (-1,-1) represents a move to the bottom left square and similarly the element (1,1) represents a move to the top right square.

At any point a player may make a statement by selecting a single element from the vocabulary.  The universe then responds to the statement.  As noted the players may make any statement at any time.  The notion of "turn" or "valid move" is not instilled in the player but in the rules of the universe.  This being the case it is possible for the players to make statements which do not change the state of the board based on the rules of the universe.  For instance, assume player 1 makes the statement (1,1) and player 2 subsequently makes the same statement.  The second statement is, in a sense, rejected by the universe as the state of the board is not changed by the statement. 

Such statements I will call meaningless.  Thus the meaningfulness of a statement is determined by whether the statement results in a change of the state of the board.  If it does, then the statement is considered meaningful, otherwise it is considered meaningless.  Of course, the rules of the universe determine whether a given statement changes the state of the board.

Given this abstraction one can conceive of a learning algorithm applied to the players such that the players learn from each statement made.  In order to learn the players must be directed toward some goal.  If they are not then the algorithm has no bearing with which to process each move.  One obvious goal of the game would be to win.  A similar and secondary goal would be to not lose.  A third rule of some usefulness would be to make only meaningful statements.

Given these goals, coupled with well defined rules over the universe, the players could conceivably "learn" the rules of tic-tac-toe based on their observations  of the reactions of the universe to their statements and the other player's statements.  Further, the players may develop stratagem coinciding with their goals of winning and not loosing (though this later item is of lesser interest to me at the moment).


DnL8Tar

-PCM